where \({\bf U}\)  is a unitary \(m\times m\)  matrix, \(\widetilde{{\bf C}}\)  is an \(m\times m\)  diagonal matrix
whose elements are given by \((\widetilde{{\bf C}})_{jj}= c_j= \lambda _j\gamma /(1+\gamma \lambda _j)\) ,
\(j=1,\ldots , m\) , with \(c_j=0\) , for \(j=n+1,\ldots ,m\) . Since we assume that \({\bf W}={\bf H}\mathsf {^H}{\bf H}\)  is
unitarily invariant, its eigenvalues do not depend on \({\bf U}\) . Moreover, \({\bf U}\)  is a Haar matrix (see
Definition REF ). Then, \(p({\bf U}|\Lambda )=p({\bf U})\) . The inner integral over
\({\bf U}\)  can be solved using the Harish-Chandra-Itzykson-Zuber integral [1]}
\( \int _{{\cal U}(m)} {\rm e}^{\mathsf {Tr}\lbrace \widetilde{{\bf C}}{\bf U}\mathsf {^H}{\bf Y}{\bf Y}\mathsf {^H}{\bf U}\rbrace }p({\bf U}) {\rm \,d}{\bf U}= \frac{\Gamma _m(m)|{\bf E}|}{\pi _m{\cal V}(\widetilde{{\bf C}}){\cal V}({\bf Y}{\bf Y}\mathsf {^H})}\,. \) 
